Time-varying optimization problems are central to many engineering applications, where performance metrics and system constraints evolve dynamically with time. Several algorithms have been proposed to address these problems; a common characteristic among them is their implicit reliance on knowledge of the optimizers' temporal variability. In this paper, we provide a fundamental characterization of this property: we show that an algorithm can track time-varying optimizers if and only if it incorporates a model of the temporal variability of the optimization problem. We refer to this concept as the internal model principle of time-varying optimization. Our analysis relies on showing that time-varying optimization problems can be recast as output regulation problems and, by using tools from center manifold theory, we establish necessary and sufficient conditions for exact asymptotic tracking. As a result, these findings enable the design of new algorithms for time-varying optimization. We demonstrate the effectiveness of the approach through numerical experiments on both synthetic problems and the dynamic traffic assignment problem from traffic control.
Internal models for nonlinear output agreement and optimal flow control
University of Stuttgart（斯图加特大学）；Institute for Systems Theory and Automatic Control, University of Stuttgart（斯图加特大学系统理论与自动控制研究所）；University of Groningen（格罗宁根大学）；ITM, Faculty of Mathematics and Natural Sciences, University of Groningen（格罗宁根大学数学与自然科学学院信息技术与建模中心）
